Let $S_0=\bigl\{(0,0,2)\bigr\}$ and for each $n\in\mathbb N$, let $S_n$ be the union of $S_{n-1}$ with the set$$\{a+1,b+1,c)\,|\,(a,b,c)\in S_{n-1}\}\cup\{(a+1,b,c+1)\,|\,(a,b,c)\in S_{n-1}\}.$$Then $S=\bigcup_{n\in\mathbb{Z}^+}S_n$. I will prove by induction that $\bigl(\forall(a,b,c)\in S_n\bigr):a+b+c$ is even.
If $n=0$, this is trivial. Pick $n\in\mathbb{N}$ and suppose that $\bigl(\forall(a,b,c)\in S_{n-1}\bigr):a+b+c$ is even. Let $(a,b,c)\in S_n$. Then there is a $(a',b',c')\in S_{n-1}$ such that $a=a'+1$, $b=b'+1$ and $c=c'$ or that $a=a'+1$, $b=b'$ and $c=c'+1$. Furthermore, since $(a',b',c')\in S_{n-1}$, $a'+b'+c'$ is even. But $a+b+c=a'+b'+c'+2$, and therefore it is even too.